VIBRATION ANALYSIS OF FLUID-STRUCTURE INTERACTION USING TUNED MASS DAMPERS

This paper has investigated the semi-analytical analysis of the solid-fl uid interaction vibration in the presence of concentrated mass-spring-damper vibration absorber. The nonlinear partial differential equations of motion are derived by considering von Karman-type large deformations and viscoelastic behaviour. Fluid-structure interaction is modelled by using an acceleration coupling model in which a nonlinear Van der Pol oscillator simulates fl uctuating nature of the vortex street. The nonlinear equations are discretized via the Galerkin approach, and the obtained equations are numerically solved by applying the Runge–Kutta method. Eventually, the dynamic response, phase plane plots, and variations of maximum amplitude in terms of fl uid velocity for different parameters are extracted. The results reveal that utilizing vibration absorber leads to a signifi cant effect on the dynamic characteristics of the system, displaces the lock-in phenomenon, and remarkably reduce the amplitude of the system oscillations.


INTRODUCTION
Fluid-induced vibration which is a nonlinear and self-excited phenomenon occurs in many engineering applications such as aircraft wing and body [1][2][3], offshore oil and gas pipelines [4][5][6], marine structures [7], yacht and ship motors [8] and biomedical engineering [9,10]. In many of these cases, this type of vibrations can cause fatigue and failure of structures [11], e.g., fatigue caused by fl uid-induced has always been a threat to aircraft fl ight. Therefore, the study of fl uid-induced vibration in structures is theoretically and experimentally considered by many researchers [12][13][14]. Bavil [13] investigates the fl uid behavior in a rectangular channel with skewed circular ribs. The operating frequency of the single-sided can be tuned similarly to a tuned mass damper (TMD). In [15] explored the application of shape memory alloy as the damping piece of pounding tuned mass dampers to mitigate the fl uctuation of suspended pipes. Studies show that if the fl uid fl ow velocity is somehow that vortex shedding frequency comes close to the fun- Figure 1: application of a dynamic vibration absorber for control of a marine shafting system [9] damental natural frequency of the system, large and severe oscillations are created in the structure. This phenomenon, known as the lock-in phenomenon creates large-scale fl uctuations, which, if not controlled, can damage the structure or even destroy it. Wu et al. [16] studied the effects of fl uid-induced vibration on structures. Kasiri Ghahi and Sanaeirad [17] studied the vibration behaviour of the cylinder at Reynolds number between 144 and 190. Their results show that the amplitude of the oscillations has a signifi cant effect on vortex formation. As the amplitude increases, the vortex lengths and the transverse distance between them decrease, and the longitudinal distance between the vortices varies inversely with the frequency of vibration. In [18][19][20] investigated the lock-in phenomenon and vortex-induced vibrations in long beams. The results of their studies showed that the lock-in area is strongly dependent on Reynolds number in long beams. Daniels et al. [21] applied numerical methods to investi- (3) (4) (5) Figure 3: Free body diagram of a beam element Gholmreza Zarepour, et al. -Vibration analasys of fl uid-structure interaction using tuned mass dampers 2 cylinders. They also studied the behaviour of the system in turbulent fl ow regions. Marra et al. [22] used empirical tests to improve the analytical model of fl uid-induced vibration in rectangular cross-section cylinders. Han et al. [23] investigated the frequency ratio on fl uid-induced vibration in cylinders. They simulated cylinder by using a fi nite element method in the Reynolds range of 200 and analyzed the transient response of the system. Jiang et al. [24] applied numerical methods to investigate fl uid-induced vibration in cylinders with a square cross-section. Vibrations caused by vortices, in which a structure interacts with a fl uid fl ow, and is exposed to vibrations perpendicular to the fl ow, can ultimately lead to fatigue and failure of the structure. Therefore, studying the applied methods for control and reduction of vibrations caused by vortices is of great importance. Investigations carried out in this fi eld show that tuned mass dampers are inexpensive and straightforward control equipment that can be used in various applications to reduce vibrations. The high performance of TMD absorbers has been reported in multiple applications such as pipelines, towers, beams, and so on. Wang et al. [25] studied the vibration behaviour of the bridge decks using a pendulum massspring-damper system. They used numerical simulations to determine the optimal absorber characteristics, and then experimentally evaluated the designed absorber performance. The results of their study showed that for the absorber with a mass ratio of 2%, the maximum amplitude of the bridge deck oscillations is reduced by 94%. Chen et al. [26] studied the vibration control in a simply supported beam under external harmonic loads, using non-linear mass-spring-damper vibration absorber. Pais and Boote [27] developments a tuned mass damper vibration absorber for yacht structures. They show that the adoption of TMD system could be a suitable solution both at the design stage when for the sake of weight containment heavy structure reinforcements should not be advisable and after construction when any other intervention is not possible.
Reviewing the conducted studies shows that control of fl uid-induced vibration in viscoelastic beams by using mass-spring-damper vibration absorber has not been studied yet. Accordingly, the present study semi-analytically investigates the fl uid-induced vibration in viscoelastic beams having TMD vibration absorbers. Nonlinear equations are derived by taking into account large deformations and the Kelvin-Voigt viscoelastic model for Euler-Bernoulli beam. After discretization of the equations using the Galerkin method, the effects of different parameters on the dynamic response of the beam is studied. Regarding the transverse vibration of the beam, for the Kelvin-Voigt damping model, which is a two-parameter rheological model, the constitutive relation is expressed as follows:

RESEARCH METHOD
In order to derive the governing differential equation of the beam, the von Kármán type nonlinear strain-displacement relationship of the beam is used. Accordingly, the nonlinear strain associated with the displacement fi eld of the transverse vibration is obtained: Considering bending stress σ x in Eq. (1), net axial force, N (x,t), and the bending moment, M (x,t), created in the viscoelastic beam is expressed as: Substituting Eq. (2) in Eq. (3), the created axial force is expressed as follows: In the following, to derive the nonlinear differential equation, free body diagram of a beam element with the length dx shown in Fig. 3 is used. Here, M(x,t), Q(x,t) and f(x,t) represent bending moment, shear force and the external force applied to the beam, respectively.
(8) Given that the inertial force of the beam element is as follows: writing Newton's second law along the vertical axis, the following equation is obtained: The moment of the forces about point P gives: By neglecting the higher powers of dx, the Eq. (8) is simplifi ed as follows: Using the last equation and Eqs. where f(x,t) is the external force applied to the beam, which is divided into two parts: the force applied by the fl uid and the pressure applied to the beam by the vibration absorber. Considering the fl uid-induced vibration, the external force applied by the fl uid consists of two parts: lift force f L and the force caused by hydrodynamic damping applied to the beam in a transverse direction f D .
According to the studies conducted by Facchinetti et al. [28] and Keber and Wiercigroch [29], these forces are expressed as follows: where C D is damping coeffi cient which depends on Reynolds number. When the Reynolds number is a subcritical range, i.e. 40<Re<30x10 5 , a low-pressure zone is created behind the structure, in which case the vortex shedding mathematical modelling shows good compatibility with experimental results [28]. In the present study, the value of C D is equal to 2.0. Lift coeffi cient C L0 is assumed to be equal to 0.3 [28]. Given that q(x,t) is lift reduced coeffi cient function, the behaviour of the wake region can be expressed as [28]: where F d is the force applied to the structure by the fl uid, and δ is the fl uid fl ow damping coeffi cient which depends on the average drag coeffi cient and is usually equal to 0.3 [30]. The vortex shedding frequency ω S depends on the fl uid velocity and Strouhal number: where S t is the Strouhal number and dependent on the Reynolds numbers range and geometry of the cylinder's cross-section, the value chosen in this study is 0.2 [30]. The effect of the vortex-induced vibration, which is represented by q(x,t) is usually considered proportional to the amplitude. According to the theory provided by Facchinetti et al. [28], the equation showing the best agreement with experimental results is as follows: where P=12 which is calculated from experimental data fi tting [28].
Having m TDM , K and C as mass, stiffness and damping of the TMD vibration absorber, using Newton's second law, the equation of TMD vibration absorber which is shown in Fig. 1, is obtained as follows: Considering Eqs. (11)- (12) and (17), the external force applied to the beam is expressed as follows: where δ(x-d) is Dirac delta function. Dimensionless variables are defi ned as: where Ω indicates dimensionless natural frequency and the parameter ε is used to demonstrate extremely small parameters. Utilizing the defi ned dimensionless vari- Gholmreza Zarepour, et al. -Vibration analasys of fl uid-structure interaction using tuned mass dampers 4 ables, the governing differential equation and the corresponding boundary conditions can be expressed as the following form: In the present study, the nonlinear PDEs presented in Eq. (20) are discretized using the Galerkin method. Accordingly, the solution of the differential equation is considered as follows: where φ n (ξ) and γ n (τ), represent the vibration mode shapes and generalized coordinates, respectively. The mode shape functions should satisfy all the boundary conditions of the system. Accordingly, in the present study, concerning the simple supports boundary conditions at the two ends, these functions are used as Considering the orthogonality of the special functions, the following conditions are satisfi ed: where δ rs is Kronecker delta function. Substituting the last equation in Eq. (20), and using the orthogonality of vibration mode shapes, the nonlinear governing equations are rewritten as: By applying the Galerkin method for Eq. (24), the following equations with ordinary derivatives are obtained: By defi ning vectors of the generalized variables q=[q 1 ,q 2 ,...,q N ] T , the Eq. (27) can be expressed in the following matrix form: The coeffi cients of Eq. (25), which are the elements of the matrices appearing in Eq. (26), are determined using the following equation: Considering the fi rst ten mode shapes in Galerkin method, i.e. N=10, the differential equations with ordinary derivatives obtained in Eq. (24), appear as 13 coupled nonlinear differential equations. Runge-Kutta numerical method is applied to solve these equations, and in the next section, the results of the numerical solution will be discussed.

RESULTS
There has been no study conducted to investigate the effects of vibration absorber on the fl uid-inductive vibration behaviour of viscoelastic beams. Therefore, in order to verify the results of the presented model, by ignoring the impact of external fl uid fl ow and vibration absorber damping, i.e., q=c=0, a similar system of simply supported beam studied by Rossi et al. [31] is taken into account.
In Table 1, the fi rst two natural frequencies of the system and the natural frequency of the TMD absorber Ω Abs are compared with the corresponding results of [31] for different dimensionless mass and stiffness parameters of the absorber. As can be seen, there is an acceptable agreement with the results of [31], and the maximum er-   Figure 6: The maximum amplitude of the system oscillations with vortex shedding frequency for different values of absorber dimensionless mass amplitude is 0.084 and 0.22, respectively. In this case, the effect of fl uid inertia can be neglected, and only the effect of the fl uid mass in reducing the natural frequency of the beam is taken into account. At low fl uid velocities, which corresponds to small Reynolds and Strouhal numbers, by increasing the fl uid velocity, Von Karman vortex streets are created symmetrically behind the beam due to negative pressure. This applies lift and drag forces to the beam, which increases the amplitude of the vibrations (Fig. 4b). Moreover, the results show that the use of TMD vibration absorber in structures which are affected by fl uid-induced vibrations causes a signifi cant reduction in the system oscillation amplitude; For instance, at fl uid velocities u=0, u=0,5 and u=1 oscillation amplitude of the system is decreased by 47%, 41% and 86%., respectively. Vibration absorber reduces the maximum amplitude of the system oscillations. For this purpose, by defi ning the parameter in Eq. (28), which represents the percentage of decrease in the amplitude of the system oscillations, the effect of vibration absorber on the behaviour of viscoelastic beam is investigated.
where y 0 max (0,5, τ) and y max (0,5, τ) indicate the maximum amplitude of the beam oscillation in the absence and presence of TMD vibration absorber. In order to study the effect of vibration absorber on the system performance, Fig. 5 shows a comparison between the vibration response of the beam in the presence and absence of the TMD vibrating absorber for different values of the absorber stiffness parameter. Here, with the constant values u=0 and εα=5x10 -3 , and also absorber parameters μ=0,2 and c=0,5, the effect of the absorber stiffness has been studied. As can be seen, the presence of the TMD vibration absorber signifi cantly reduces the amplitude of the system oscillations. The decrease depends on the absorber parameters; by increasing k, the effect of absorber on decreasing the amplitude of the oscillations is lowered. As an example, for k=500 and k=1, the decrease in the amplitude of the system oscillation are 13.6% and 31.5%, respectively. According to the results, it is observed that absorber with stiffness k=10, which reduces the amplitude of system oscillations by 75%, has the best performance. Considering that in this case, the dimensionless natural frequency of the absorber Ω abs =√k/μ=7,07 is near the natural frequency of the beam with u=2, the TMD system absorbs the highest amount of energy, and reduction of the amplitude of the oscillations has the maximum value. In order to investigate the dynamic behaviour of the viscoelastic beam with TMD vibration absorber under the fl uid fl ow excitation, the frequency response of the system is extracted in the steady-state. Fig. 6 demonstrates the maximum amplitude of the system oscillations with vortex shedding frequency for different values of absorbent dimensionless mass. As the results show, the jump phenomenon occurs due to the nonlinear behaviour of the system. In the absence of the TMD vibration absorber, increasing the parameter Ω S , resonance occurs in the system. Resonances occur at Ω S about 3.6 and 5.3, and when it reaches 13.5, jump phenomenon occurs, and the system behaviour becomes unstable. Another result from Fig. 6 is that the use of vibration absorber has a signifi cant effect on the frequency-response curve of the system, and by increasing the absorber dimensionless mass parameter, jump phenomenon is eliminated from the system behaviour. For β=0,1, jump phenomenon is seen at Ω S =12,3; increasing the Ω S maximum amplitude of the oscillations is sharply reduced, and the resonance regions omitted from the system behaviour. These results show the useful ability of the presented method to control vibrations in viscoelastic beams affected by external fl uid fl ow.
To study the effect of fl uid fl ow on the dynamic response of the system in stable and unstable conditions, the steady-state time response and also phase and Poincaré diagrams for the viscoelastic beam without vibration absorber are shown in Figs. 7 to 9. In Fig. 7, which is plotted for the conditions of the point A (see in Fig. 6), the vibration amplitude is stable, and the time response is periodic so that the Poincaré map shows a point in the phase plane. Near point B, subharmonic resonances occur in the system, the vibration amplitude is unstable and has a semi-periodic cycle which is visible in Fig. 8. Such a behaviour is also observed for the system in the region C in Fig. 9.

CONCLUSION
In this study, the ability to utilize mass-spring-damper vibration absorber focused on controlling fl uid-induced vibration in viscoelastic beams was semi-analytically investigated. Nonlinear equations governing the motion were derived by taking into account the effects of nonlinear strains and were discretized by Galerkin's method.
The results show that applying the vibration absorber signifi cantly reduces the amplitude of the system oscillations, i.e., the maximum amplitude is decreased at fl uid velocities u=0, u=0,5 and u=1 by 47%, 41% and 86%, respectively. Concerning the infl uences of the TMD system stiffness on the responses of the structure, the results showed that increasing the stiffness of TMD; the energy pumping effect can be enhanced with the presence of stable responses, thereby reducing the vibration ampli-